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I 


INFINITESIMALS 
AND  LIMITS. 


BY 

JOSEPH  JOHNSTON  HARDY, 

PROFESSOR  IN  LAFAYETTE  COLLEGE. 


EASTON.  PA. : 
THE  CHEMICAL  PUBLISHING  CO. 

1912. 


INFINITESIMALS 
AND  LIMITS. 


BY 


JOSEPH  JOHNSTON  HARDY, 

PROFESSOR  IN  LAFAYETTE  COLLEGE. 


EASTON.  PA. : 

THE  CHEMICAL  PUBLISHING  CO. 

1912. 


Copyright,  1900,  by  Edward  Hart. 


•  •  .•  •    • 

•  •  *  •  •    • 

•  •    •    •• 


:3S 


INFINITESIMALS  AND  LIMITS 


CHAPTER  I 
INFINITESIMALS 


/='' 


^o 


Fig.  I. 


1.  In  Fig.  I  let  AB  and  CD  be  two  parallel  straight 
lines  and  draw  EF  J_  CD.  I,et  a  point  P  start  from  E 
and  move  along  AB  to  the  right  so  as  at  the  end  of  the  first 
second  to  be  at  P,  at  the  end  of  the  second  second  to  be 
at  P',  at  the  end  of  the  third  second  to  be  at  P", 

As  the  point  P  changes  its  position, 
Z.  EFP  takes  the  different  values  EFP,  EFP',  EFP",  etc. 
Z  PFD         ''  "  PFD,  P'FD,  P"FD,  etc. 

EP        ''  ''  EP,     EP',     EP",  etc. 

FP         •'  '*  FP,      FF,     FP",  etc. 

but  ^^  EFD  does  not  change  its  value, 
and      EF 

That  is,  among  the  quantities  which  enter  into  the  dis- 
cussion of  the  given  figure  as  the  point  P  changes  its  posi- 
tion, there  are  some,  Z.  EFD  and  EF,  which  retain  the 
same  values,  and  which  are  accordingly  called  constants. 


4 


'S  r>  /»  ^'^  *> 


4  INFINITESIMALS   AND   I^IMITS 

Others,  as  EP,  Z!  EFP,  ^  PFD  and  FP,  which  assume 
different  values,  are  called  variables.  . 

/}  — — , , , — >  a 

Fig.  2. 

Again,  suppose  a  man  starts  from  A  to  go  to  B,  going 
over  half  the  distance  between  himself  and  B  every  hour. 
At  the  end  of  the  first  hour  he  will  reach  M  ;  at  the  end 
of  the  second  he  will  reach  M' ;  at  the  end  of  the  third  at 
M";  etc. 

In  this  illustration  we  see  that  AM,  the  distance  the 
man  travels,  and  MB,  the  distance  from  the  man  to  B, 
change  continually  ;  but  no  matter  how  long  the  man 
travels,  AB  always  retains  the  same  value.  Hence  we 
call  AM  and  MB  variables  and  AB  a  constant. 

In  the  series 

I  4-  >^  +  ^  -f  >^  +  etc., 
let  n  be  the  number  of  terms  taken  beginning  with  the 
first  and  let  s  be  the  sum  of  those  terms. 

Now  let  n  be  made  greater  and  greater  continually. 
Then  when  «=2,     s=i  -{-  }4  =  i}4 

«=4,   s=i-^  y2 -h  }i -\- 'A         =^V^ 

We  see  from  these  equations  that  n  and  s  change  their 
values  continually  but  that  each  term  of  the  series  always 
retains  the  same  value. 

Hence  we  call  n  and  s  variables  and  each  term  a 
constant. 

Thus  in  any  problem  there  may  appear  quantities 
which,  under  the  assumptions  made  in  the  problem, 
always  retain  the  same  values.     There  may  also  appear 


INFINITESIMALS   AND   LIMITS  5 

Others  which  under  the  same  assumptions  take  different 
values.  The  former  are  called  constants,  the  latter 
variables. 

2.  A  variable  is  a  quantity  which  changes  its  value 
under  the  assumptions  made  in  the  problem  into  which  it 
enters. 

3.  A  constant  is  a  quantity  which  does  not  change  its 
value  under  the  assumptions  made  in  the  problem  into 
which  it  enters. 

4.  The  absolute  value  of  a  number  is  simply  the  num- 
ber of  units  it  contains,  no  regard  being  paid  to  its  sign. 

Thus  the  absolute  value  —  5  and  of  5  is  5. 

5.  An  infinitesimal  is  a  variable  which  approaches  zero 
in  such  a  way  that  its  absolute  value  may  be  made  to 
become  and  remain  less  than  any  positive  number  c  that 
may  be  named,  however  small  the  e  may  be  taken. 


In  Fig.  3  let  the  point  P'  remain  fixed,  let  P  move  con- 
tinually along  the  curve  till  it  reaches  P',  and  let  it  stop 
there. 

Then  the  ^  c  continually  approaches  zero  in  such  a  way 


6  INFINITESIMALS    AND    LIMITS 

that  its  absolute  value  may  be  made  to  become  and 
remain  less  than  any  positive  number  c  however  small  e 
may  be  taken. 

For  let  €  =  1°. 

Then  by  moving  P  far  enough  we  can  make 

c<  1°. 
Now  let  €  =  i'. 

By  moving  P  farther  on  we  can  make 

e<  i'. 
Let  e  =  i". 

Again  by  moving  P  farther  on  we  can  make 

c  <  i". 

Similarly  no  matter  how  small  €  is  taken  we  can  make 

C<e 
by  moving  P  on  far  enough. 

Since  by  hypothesis  P  continually  approaches  P'  and 
does  not  pass  it,  c  continually  approaches  o  but  cannot 
become  less  than  o.  Therefore  when  the  absolute  value 
of  c  once  becomes  less  than  e  it  must  remain  so. 

Hence  by  the  definition  c  is  an  infinitesimal. 

Again  in  Fig.  3  the  line  PK  continually  approaches 
zero  in  such  a  way  that  its  absolute  value  may  be  made 
to  become  and  remain  less  than  any  positive  number  e, 
however  small  the  e  may  be  made. 

Let  €=.i. 

Then  by  moving  P  towards  P'  we  can  make 

PK  <  .1. 

Now  let  €  =  .01. 

Then  by  moving  P  further  on  we  can  make 
PK  <  .01. 


INFINITESIMALS    AND    LIMITS  ^ 

Similarly,  no  matter  how  small  e  may  be  taken  we  can 
make  PK  <  e 

by  moving  P  far  enough. 

Since  by  hypothesis  P  continually  approaches  P'  and 
cannot  pass  it  the  absolute  value  of  PK  continually 
approaches  o  and  cannot  become  less  than  o.  Therefore 
when  the  absolute  value  of  PK  once  becomes  less  than 
€  it  must  remain  so. 

Hence  by  the  definition  PK  is  an  infinitesimal. 

Similarly  it  may  be  shown  that  the  A  PKP'  is  an 
infinitesimal. 

It  is  obvious  that  each  of  these  infinitesimals  c,  PK 
and  A  PKP'  will  become  zero  when  P  reaches  P'. 

In  Fig.  4  let  AB  and  CD  be  two  parallel  straight  lines. 
Draw  P"K  J_  CD. 


C 

Fig.  4. 

Now  let  B  remain  fixed  and  let  P  move  continually 
along  AB  towards  the  right. 

The  Zl  PHD  continually  approaches  zero  and  its  abso- 
lute value  may  be  made  to  become  and  remain  less  than 
any  positive  number  e,  how^ever  small  e  may  be  made. 

Hence  by  the  definition  ^  PED  is  an  infinitesimal. 

It  is  obvious  that  Zl  PED  can  never  be  made  equal  to 
zero. 


8  INFINITESIMALS   AND   LIMITS 

IvCt  5  =  the  sum  of  72  terms  of  the  series 
I  -V  %  ^Va  -^  }i^  etc. 
I.et  R  =  2  —  s 

Now  let  n  increase  continually.  Then  s  will  increase 
continually  and  can  be  made  as  nearly  equal  to  2  as  we 
please.  Hence  R  will  continually  approach  zero  and 
may  be  made  less  than  a  positive  number  e,  however 
small  c  may  be  taken. 

Since  all  the  terms  are  positive,  R  must  be  positive  ; 
hence  when  once  R  becomes  less  than  c,  it  must  remain 
so. 

Hence  by  the  definition  R  is  an  infinitesimal. 

It  is  obvious  that  R  can  never  be  made  equal  to  zero. 

6.  There  are  therefore  two  classes  of  infinitesimals, 
namely  : 

( 1 )  Those  which  may  finally  become  zero. 

(2)  Those  which  can  never  become  zero. 


CHAPTER  II 
INFINITES 

7.  An  infinite. — An  infinite  is  a  variable  whose  abso- 
lute value  can  be  made  to  become  and  remain  larger  than 
any  positive  number  that  may  be  assigned. 

In  Fig.  4  the  line  P"P  is  an  infinite  ;  for  by  moving  P 
farther  and  farther  to  the  right  we  can  make  the  absolute 
value  of  P"P  greater  than  any  positive  number  that  may 
be  taken.  Also  since  the  absolute  value  of  P"P  is 
always  increasing,  when  once  it  becomes  greater  than  any 
positive  number,  it  will  remain  greater. 


.  INFINITESIMAI.S   AND    I.IMITS  9 

Again  let 

[i]  5=i-i-2  +  44-  etc.  to  n  terms. 

Then  5  is  an  infinite. 

For   the  series  is  a  Geometrical  Progression  and  we 

know  by  arithmetic  that 

ar—  a 
[2]  s  = 

2"  —  I 
[3]  .-.  ^=- ^  =  2"  — I. 

L^J  2—1 

Now  we  see  from  the  right-hand  side  of  [3]  that  by 
making  n  larger  and  larger  we  can  make  s  larger  than 
any  positive  number  that  may  be  taken.  Also  since  s 
is  always  increasing  when  once  it  becomes  larger  than 
any  positive  number,  it  will  remain  larger. 

The  symbol  for  an  infinite  is  00 . 

8.  A  finite. — Any  quantity  which  is  neither  an  infini- 
tesimal nor  an  infinite  is  called  a  finite. 


CHAPTER  III 

PROPERTIES  OF  INFINITESIMALS 

9.  In  Fig.  5  draw  PQ,  RS,  etc.,  J_  xx'  and  at  equal 
distances  from  each  other.     Also  draw  RT,  etc. ,  parallel 

to  xx\  .  , 

We  will  in  this  way  construct  the  triangular  figure 
PRT  bounded  by  PT,  RT  and  the  curve  ;  also  the  trian- 
gular figures  a,  b,  c,  and  d ;  and  the  rectangles  q,  p,  and  o. 

Now  let  PQ  move  continually  towards  P'Q',  the  other 
perpendiculars  moving   so  as   always   to   preserve   the 


lO 


INFINITESIMALS   AND   LIMITS 


same  relative  position.     Then  it  is  obvious  that  PP'Q'Q. 
^,  /,  o,  a,  b,  c,  d  are  all  infinitesimals.  by  §  5. 


Fig.  5- 

We  see  also  that    q  <p  <o  <^  PP'Q'Q. 

Hence  one  infinitesimal  may  be  greater  than  another. 

Also  since 
a  ^  b  -^  c-^d  -^  etc.  -V  q  ^  p  ^  0  ^  etc.  =  PP'Q'Q. 
it  is  obvious  that  the  sum  of  a  finite  number  of  infinitesi- 
mals may  be  an  infinitesimal. 

Finally  let  there  be  any  n  -j-  i  infinitesimals  like  q,  p,o. 
Then  since  q  <i  P  <^  0  ... 

p-\-  0^-    ... 


we  get 
Let 
Then 


q  < 
p-ho 


.  =  J 


?< 


But  q  and  5  are  both  infinitesimals. 


INFINITESIMAI.S   AND   I.IMITS  II 

Hence  an  infinitesimal  may  be  less  than  another 
infinitesimal  divided  by  a  constant. 

It  is  obvious  from  this  discussion  that  an  infinitesimal 
can  be  properly  said  to  be  very  small  only  at  certain 
stages  of  its  variation. 

When  a  variable  becomes  smaller  and  smaller  in  such 
a  way  that  it  may  be  made  smaller  than  any  positive  num- 
ber that  may  be  named,  it  is  said  to  decrease  indefinitely. 

When  a  variable  becomes  larger  and  larger  in  such  a 
way  that  it  may  be  made  larger  than  any  positive  number 
that  may  be  named,  it  is  said  to  increase  indefinitely. 

PROPOSITION    I 

10.  The  reciprocal  of  an  infinitesimal  is  an  infinite. 
For  let  a  ^  any  infinitesimal. 

Then  —  =  its  reciprocal, 

a 

Now  when  a  decreases  indefinitely, 

—  increases  indefinitely  ; 

a 

for  when  the  denominator  of  a  fraction  decreases  indefi- 
nitely the  value  of  the  fraction  increases  indefinitely. 

Hence  —  is  an  infinite.  by  §  7. 

a 

Q.    E.    D. 

PROPOSITION    2 

11.  The  product  of  a  constant  a  nd  an  infinitesimal  is  itself 
an  infinitesimal. 

Let  €  and  s  =  two  infinitesimals 

and  n  =  any  constant. 

Then  we  may  take  5  <  — .  by  §  5. 

n 

Hence  ns  <  c  Q.  K.  D. 


12  INFINITESIMALS   AND    LIMITS 

PROPOSITION   3 

12.  The  sum  of  a7iy  finite  number  of  infinitesimals  is 
also  an  infinitesimal . 

Let  there  be  n  infinitesimals. 

Let  c^  be  that  one  of  these  infinitesimals  which  has  at 
any  instant  the  greatest  absolute  value. 

[i]   Let  J  =  €,  -i-  e,  +  €3  +  . .  .  H-  6„. 

[2]  Then  s  <  ne^, 

But  nt^  is  an  infinitesimal.  by  §   11. 

Hence  by  [2]  ^  is  also  an  infinitesimal.  q.  e.  d. 

PROPOSITION   4 

13.  The  product  of  any  finite  number  of  infinitesimals  is 
also  an  infinitesimal. 

Let  there  be  n  infinitesimals 

^1.    €2.    «3.       •    ••.    €„. 

Let  ;r  be  a  variable  which  can  be  made  to  increase  in- 
definitely.    We  can  alwa3^s  find  a  value  of  x  such  that 

by  §  5. 


[i]   Hence 


^I< 

I 
X   ' 

e.< 

I 
X   ' 

^3< 

I 

X  ' 

etc  <  etc., 

^«< 

I 

X  ' 

Cj   €,  €3    .... 

€„< 

I 
x„ 

INFINITESIMAI.S   AND    LIMITS  1 3 

Now  making  x  increase  indefinitely  we  can  make  —  an 

infinitesimal.     Hence  by  [i]   c^  f..^  S   •  •  •   ^n  is  an   infin- 
itesimal. Q.  E.   D. 
PROPOSITION   5 

14.  First. — A  power  of  an  infinitesimal  is  itself  an 
infinitesimal  if  its  exp  orient  is  a  positive  finite  other  than  o. 

Second. — //  is  an  infinite  if  its  exponent  is  a  negative 
finite  other  than  o. 

Third. — //  is  equal  to  i  if  its  exponent  is  o. 

First. 

Let  a  be  any  infinitesimal  and  p  a  positive  finite  other 
than  o. 

[i]  Then  a^  =  a  a  a  a  ...   to  /  factors. 

[2]   But  a  a  a   ...   to  /   factors  is  an   infinitesi- 

mal, by  §   13. 

[3]   Hence  a^  ^^  an  infinitesimal. 

Second. 
Let  —  /  be  a  negative  finite  other  than  o. 

[4]  Then  ''~'=^' 

But  since  a  is  an  infinitesimal 

[5]  a^  =  an  infinitesimal,  by  case  i. 

by  §   10. 


[6]  and 

I 

=  an 

infinite. 

Hence  by  [4] 

we 

get 

[7] 

a^ 

—  an 
Third. 

infinite. 

Let 

P 

=  0. 

Then 

a.'  . 

=  a«  = 

=  I. 

Q.    E.    D. 


H 


INFINITESIMALS   AND   LIMITS 

CHAPTER  IV 


LIMITS 
15.  Limit. — The  limit  of  a  variable  is  that  constant 
which  differs  from  the  variable  by  an  infinitesimal. 


;■•  •     ' 

x:^ 

<- 

f. 

rt 

_yyi 

-n 

r 

^ 

/ 

ti 

Fig.  6. 


h 


In  Fig.   6 

[i]  ^^a=  Z.b  -\-  Z.C        by  Geom. 

[2]   .-.  ,^a  —  Z.b  =  ^^c 

Now  let  P  move  along  the  curve  till  it  reaches  P'. 

Then  ^^i  is  a  variable,  z^  <^  is  a  constant  and  ^cis  an 
infinitesimal. 

Hence  by  [2]  Zl  ^  is  that  constant  which  differs  from 
the  variable  z!l  a;  by  an  infinitesimal  ^  c. 

Therefore  by  the  definition  Zl  ^  is  the  limit  oi  ^  a. 

Again  ^ 

[3]  PH  —  P'H'  =  PK. 

Now  again  let  P  move  along  the  curve  till  it  reaches  P'. 
Then  PH  is  a  variable,  P'H'  is  a  constant,  and  PK  is  an 
infinitesimal. 

Hence  by  [3]  P'H'  is  that  constant  which  differs  from 
the  variable  PH  by  the  infinitesimal  PK. 

Therefore  by  the  definition  P'H'  is  the  limit  of  PH. 

It  is  obvious  that  both  the  variable  ^  a  and  the  line 
PH  reach  their  limits. 


INFINITESIMALS  AND   LIMITS  1 5 

In  Fig.  4, 

[4]  Z.  P"KD  —  P"EP  -=  Z  PED. 

Let  P  move  continually  to  the  right.  Then  Z-  P"KP 
is  a  variable,  Z.  P"ED  is  a  constant  and  Z  PED  is  an  in- 
finitesimal. 

Hence  by  \_\\^ Z  P"ED  is  that  constant  which  difiers 
from  the  variable  Z  P"EP  by  the  infinitesimal  Z  PED. 

Therefore  by  the  definition  Z  P"ED  is  the  limit  of 
Z  P"EP. 

Let  Sn  =  the  sum  of « terms  of  the  series  i  +  >^  H-  /4^  etc. 

Let  R„  =  2  —  j„. 

Let  n  increase  indefinitely.  Then  5  is  a  variable,  2  is 
a  constant,  and  R«  is  an  infinitesimal.  by  §  5. 

Hence  2  is  that  constant  which  differs  from  the  vari- 
able s,,  by  the  infinitesimal  R„. 

Therefore  by  the  definition  2  is  the  limit  of  s„. 

It  is  obvious  that  neither  of  these  last  two  variables 
can  ever  reach  its  limit. 

16.  Hence  there  are  two  classes  of  Limits  : 

[i]  Those  to  which  the  variable  finally  becomes  equal. 
[2]   Those  to  which  the  variabl  e  can  never  become  equal . 
The  first  class  are  called  Attainable  Limits. 
The  second  class  are  called  Unattainable  Limits. 

17.  In  Fig  4, 

let  V  =  the  variable  Z  P"EP, 
let  /  ^  its  limit  Z  P"ED, 
and  e  =  the  infinitesimal  Z  PED. 
[i]  Since  Z  P"EP  =  Z  P"ED  —  Z  PED.     by  Geom. 
[2]  V  ^=  h  —  c 

And  in  general  whatever  the  variable  may  be, 


1 6  INFINITESIMALS   AND   LIMITS 

[3]  2;  =  /  —  £. 

In  [3]  €  may  be  either  positive  or  negative.  For  if  as 
in  Fig.  4  the  variable  always  increases  as  it  approaches 
its  limit  €  will  be  negative,  but  if  in  any  case  the  vari- 
able always  decreases  as  it  approaches  its  limit  c  will  be 
positive.  There  are  some  cases  in  which  the  variable 
alternately  passes  from  one  side  of  its  limit  to  the  other 
as  it  approaches  it.  In  these  cases  e  is  alternately  posi- 
tive and  negative. 

PROPOSITION    I 

18.  If  two  variables  are  always  equal,  and  each 
approaches  a  limits  their  limits  are  equal. 

Let  V  and  v^  be  two  variables, 

/    "     /' be  their  respective  limits, 

and  €    *'     c'  be  two  infinitesimals. 

Let  V  =^  v' 

We  are  to  prove  that  I  =^  I'. 

[i]  V  =  I  —  €  by  §   17. 

[2]  and  z/'=/' — e'.  by  §   17. 

Subtracting  [2]  from  [i]  we  get 

[3]  V  —  v'  =  l~l'  —€^  €! 

[4]   But  by  hypothesis  v  =  v\  and  hence  v  —  v^  ^=  o. 

Substituting  this  value  oi  v  —  v'  into  [3] 

[5]  we  get  o  =  /  —  /'  —  €  -{-  c'. 

[6]   Hence  /'  —  /=€'—£. 

If  €  and  e'  are  not  equal  to  each  other  suppose  that 
the  absolute  value  of  c  >  the  absolute  value  of  e'. 

[7]  Then  c'  —  e  is  not  greater  than  2€, 

and  from  [6]  it  follows  that 


I 


INFINITESIMALS   AND   LIMITS  1 7 

[8]  /'  —  /is  not  greater  than  2c. 

Since  by  §  15  /  and  V  are  constants,  /'  —  /  is  a  constant. 
And,  since  by  §  5  e  can  be  made  as  nearly  zero  as  we 
please,  2c  can  also  be  made  as  nearly  zero  as  we  please. 
Then  it  follows  from  [8]  that  V  —  /  is  a  constant  which 
can  never  be  greater  than  2c.  But  the  only  such  constant 
is  zero. 

[9]   Hence  V  ~  I  =  o. 

[10]   or  /'  —  /.  Q.   K.  D. 

PROPOSITION    2 

19.  If  the  sum  of  any  finite  number  of  variables  be 
variable,  then  the  limit  of  their  sum  is  equal  to  the  sum  of 
their  limits. 

Let  V  and  v^  =  two  variables. 

/    "     /'  =  their  respective  limits. 

and  e    "     c' =  two  infinitesimals. 

We  are  to  prove  that  lim  (z;4-2;' )=/-(-/'. 

[i]  7;  =  /— c  by  §   17,  [3] 

[2]   and  v'=l'-^  by  §   17,  [3] 

[3]   By  addition  2;  -f-  z^'  =  /  -f  /'  —  e  —  c' 

[4]   and  limit  (z;  -f  z;')  =  limit  (/-{-/'  —  c  —  c'). 

by  §   18. 

Since  by  hypothesis  e  and  e'  are  infinitesimals, 
(  —  c  —  c')  is  an  infinitesimal  by  §  12. 

Also  since  by  §  15  /  and  /'  are  both  constants,  /  +  /'  is 
a  constant. 

Hence  /  +  /'  is  a  constant  which  differs  from  the  vari- 
able [/'  4-  /  —  e  —  e']  by  the  infinitesimal  [  —  c  =  c']. 
[5]  Therefore         limit  (/-(-/'  —  £—€')  =  /-[-  /' 

by  §  15. 
Substituting  into  [4]  we  get 


1 8  INFINITESIMALS   AND    LIMITS 

[6]  limit  {v  -\-  v')  =  I  ^  I'. 

Q.    E.    D. 

Similarly  the  theorem  may  be  proved  for  the  sum  of 
any  finite  number  of  variables  since  (e  —  e'  —  e"  .  .  .  etc.) 
is  an  infinitesimal.  by  §   12. 

PROPOSITION   3 

20.  If  the  product  of  a  finite  number  of  variables  be 
variable,  then  the  limit  of  their  product  is  the  product  of 
their  limits. 

Let  V  and  v'  =  two  variables, 

/    "     /'  =  their  limits, 

and  €    *'     e' =  two  infinitesimals. 

We  are  to  prove  that  lim  vv^  =  //'. 

[i]  v=  I  —  €  by  §   17,  [3] 

[2]  v'=l'=€  ''      *' 

Multiplying  [i]  by  [2]  we  get 

[3]  z;z/  =  //'  — €/'  —  €7  4-   ec' 

[4]  Therefore 

limit  {vv')  =  limit  (//'  —  c/'  —  eV  +  «V). 

by  §   18. 

Now  €/'  and  c7  are  infinitesimals  by  §   11. 

and  ee'  is  an  infinitesimal.  by  §   13. 

Hence  —  d'  —  eV  -f  ee'  is  an  infinitesimal,  by  §   12. 

But  since  by  §   15  /  and  /'  are  constants //'  is  a  constant. 

Now  //'  is  a  constant  which  differs  from  the  variable 
ir  —  e^'  —  eV  -{-  ee'  by  the  infinitesimal  —  d'  —  e7  +  ee'. 

[5]   Hence     lim  (//'  +  «/'  — ^7 -f- ee')  = //'     by  §   15. 

[6]  Hence  by  [4]  lim  {vv')  =^  II'  q.  E.  d. 

Similarly  the  proposition  may  be  proved  for  the  product 
of  any  finite  number  of  variables. 


INFINITESIMALS   AND    LIMITS  1 9 

21.  Corollary  i. —  The  limit  of  the  product  of  a  constant 
and  a  variable  is  the  product  of  the  constant  and  the  limit 
of  the  variable. 

Let  a  =  any  constant 
"    z/  =  any  variable 
and  /  ^  its  limit 
Let  e  =  an  infinitesimal, 
[i]  Then  v  =  l—€  by    §17,  [3] 

[2]  av=  al  —  a€ 

[3]   Hence     lim  (av)  =  lim  {al  —  ae).        by  §   18. 
Now  a€  is  an  infinitesimal  by  §   11. 

but  since  by  §   15  /  is  a  constant  al  is  a  constant. 

Hence  al  is  a  constant  which  differs  from  the  variable. 
(al  —  ae)  by  the  infinitesimal  ac. 
Therefore  lim  (al  —  ae)  =  al  by  §   15. 

Substituting  into  (3)  we  get 
lim  {av)  =  al. 

Q.   E.   D. 

22.  Corollary  2. — If  the  product  of  any  finite  number  of 
variables  be  a  constant  the  limit  of  their  product  is  the  same 
constant. 

Let  v,w,x,  .  .  .  etc.  be  variables  and  let  «  be  a  constant, 
[i]  Let  vwx  . .  .  etc.  =  a. 

We  are  to  prove  that 

limit  {ywx  .  .  .  etc.)  =  a 

Let  2  be  another  variable 
[2]   Then  by  (i)  (vwx  . .  .  etc.)  z  ^=  az 

[3]  lim  (vwx  . .  .  etc.)    lim  z  =^  a  lim  js 

by  §  §  18  and  21. 
[4]  .*.     lim  (vwx  .  .  .  etc.)  =  a 

Q.   E.  D. 


20  INFINITESIMALS   AND   LIMITS 

PROPOSITION  4 

23.  The  limit  of  any  positive  integral  power  of  a  vari- 
able is  the  same  power  of  the  limit  of  the  variable. 

Let  V  =  the  variable, 

/  =  its  limit, 
and ;?  =  any  positive  integral  exponent. 
We  are  to  prove  that  lim  v^  =  V 

[i]   limit  {yvvv  ...  \.o  n  factors)  =  /././.  ...  \.o  n  fac- 
tors, by  §  30. 
[2]  Therefore        limit  if'  =  I". 

Q.   E.   D. 

PROPOSITION   5 

24.  The  limit  of  the  quotient  of  two  variables  is  the  quo- 
tient of  their  limits^  provided  that  neither  of  the  limits  be  o. 

Let  X  and  y  be  the  two  variables. 


we  are  to  pre 

>ve  tnai  nm  —   =  r^ 

y          \\my 

[I] 

Letz/=   -^ 

y 

[2]  then 

vy  =^  X 

[3]  and 

lim  {vy^)  =  lim  x. 

by  §   18. 

[4]  But 

lim  {vy)  =lim  vVimy 

by  §  20. 

[5]  hence 

lim  V  lim  y  =  lim  x 

[6]  and 

,.             lim;t 

lim  V  =  7: 

lim  y 

[7]   or  by  [I] 

,,       X         lim  x 

lim  —  =  r- • 

1/                 lim    1/ 

Q.    E.    D. 


INFINITESIMALS   AND    LIMITS  21 

25.  Corollary, —  The  limit  of  the  quotient  of  a  constant 
by  a  variable  is  the  constant  divided  by  the  limit  of  the 
variable. 

Let  a  ^  any  constant, 
«<  2,   ^  *<     variable, 
and  /  =  its  limit. 
We  are  to  prove  that 

[i]   Let 

[2]   Then 

[3]   Hence  lim  zv  =  a  by  §  22. 

[4]   and  lim  2  lim  v  ^=  a  by  §  20. 

[5]   hence 

[6]  or  by   [i]  lim  -^  ==  ^r:^ — 


V 

lim  V 

2  = 

a 

V 

2V  = 

a 

lim  2v  = 

a 

im  2  lim  v  = 

a 

lim  2  = 

a 

\\mv 

I1  fr\    — 

a 

Q.    E.    D. 


PROPOSITION   6 


26.   The  limit  of  the  nth  root  of  a  variable  ts  equal  to  the 
nth  root  of  the  limit  of  that  variable. 

Let  V  =  any  variable 

and  /  =  its  limit. 

n  n  

[i]  z/ =  y^z;"    and  /=  y^'/", 

[2]  also  limit  v  =  I  by  hyp. 

Substituting  in   this  equation  the  values  of  /  and  v 
found  in  [i]  we  get 

[4]  limit  \/lF  ==.  ylF 


22  INFINITESIMALS   AND   LIMITS 

Now  since  v  represents  any  variable  whatever,  v''  rep- 
resents any  variable  whatever  and  /"  is  its  limit. 

by  §  23. 
Therefore   [4]   shows  that  the  limit  of  the  «th  root  of 
any  variable  is  equal  to  the  nih  root  of  its  limit. 

Q.   E.   D. 


14  DAY  USE 

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